Performance of the EOMIP-CCSD (2) Method for Determining the

Article pubs.acs.org/JCTC

Performance of the EOMIP-CCSD(2) Method for Determining the Structure and Properties of Doublet Radicals: A Benchmark Investigation Achintya Kumar Dutta, Nayana Vaval, and Sourav Pal* Physical Chemistry Division, CSIR-National Chemical Laboratory, Pune-411008, India S Supporting Information *

ABSTRACT: We present a benchmark study on the performance of the EOMIP-CCSD(2) method for computation of structure and properties of doublet radicals. The EOMIP-CCSD(2) method is a second-order approximation to the standard EOMIP-CCSD method. By retaining the black box nature of the standard EOMIP-CCSD method and adding favorable N5 scaling, the EOMIP-CCSD(2) method can become the method of choice for predicting the structure and spectroscopic properties of large doublet radicals. The EOMIP-CCSD(2) method overcomes the typical problems associated with the standard single reference ab initio treatment of doublet radicals. We compare our results for geometries and harmonic vibrational frequencies with those obtained using the standard EOMIP-CCSD method, as well as unrestricted Hartree−Fock (UHF)- and restricted open-shell Hartree−Fock (ROHF)-based single-reference coupled-cluster and second order many-body perturbation theory (MBPT(2)) methods. The effect of the basis set on the quality of the results has been studied using a hierarchy of Dunning’s correlation-consistent aug-cc-pVXZ (X = D, T, Q) basis sets. Numerical results show that the EOMIP-CCSD(2) method, despite its N5 scaling, gives better agreement with experimental results, compared to the UHF- and ROHF-based MBPT(2), as well as the single-reference coupled-cluster methods.

1. INTRODUCTION In recent times, ab initio quantum chemistry has become the trusted companion of researchers for the elucidation of structures and properties of complicated molecules. Among the plethora of methods available, the single-reference coupled-cluster method,1,2 because of its systematic treatment of electron correlation, has emerged as the method of choice for accurate prediction of structure,3−5 properties,6 and vibrational spectra7−9 of closedshell molecules. The extension of the single-reference coupledcluster method to open-shell systems, based on unrestricted Hartree−Fock (UHF)10 or restricted open-shell Hartree−Fock (ROHF)11,12 references, has also been achieved. However, the single-reference coupled-cluster method, even in the singles and doubles approximation, scales as N6 power with the basis set, whereas the inclusion of partial triples13−15 (CCSD(T)) increases the scaling to N7. The development of parallel codes and a rapid increase in computational power, in recent times, may have made coupled-cluster calculations for small- and mediumsized molecules possible (at least in small basis sets). However, the structure and property calculations of medium-sized molecules, in moderately sized or big basis sets, are still not routine. Moreover, N6 scaling is computationally demanding to use in quantum molecular dynamics calculations, which involves multiple force constant calculations and is essential for predicting the time evolution and temperature effect on the molecules. Thus, a theoretical method with scaling lower than N6, that is still © 2013 American Chemical Society

able to predict the properties of the molecule accurately, is the need of the day. The standard second order many-body perturbation theory (MBPT(2)) method offers the first correlation correction to the energy over the Hartree−Fock method and scales as N5 with the basis set. Over the years, the method has been extensively used as an accurate tool for ab initio investigation on a large variety of closed-shell molecules. But, standard UHFbased16 or ROHF-based17,18 MBPT(2) methods perform poorly in the case of open-shell doublet radicals, because of multiple problems19−21 associated with the open-shell single reference wave function, such as spin contamination,22,23 symmetry breaking,24 near-singularities of the HF solution,25,26 pseudo-Jahn−Teller effects,27 and the presence of multireference character. However, ab initio investigation of doublet radicals is extremely important, because of the key role of the radicals in biology and chemistry. The high energy and short lifetime of a doublet radical makes experimental characterization often tedious, and sometimes impossible. Theoretical calculations28−34 can be helpful to understand the doublet radicals and their mechanism in chemistry and biology. However, this is more difficult for open-shell molecules, mainly because of two reasons: Received: April 18, 2013 Published: August 20, 2013 4313

dx.doi.org/10.1021/ct400316m | J. Chem. Theory Comput. 2013, 9, 4313−4331

Journal of Chemical Theory and Computation

Article

These amplitudes are generally obtained by the iterative solution of a system of coupled nonlinear equations. Extending T up to the nth order, where n equals the total number of electrons in the system, leads to the full CI solution. However, for practical applications, T is truncated to finite order. The exponential structure of the correlation operator ensures the size extensivity, even at the truncated level of T. The truncation of T amplitudes at T1 and T2 leads to the popular coupled-cluster singles and doubles (CCSD) approximation, which scales as N6 with the basis set (where N is the number of basis functions). The inclusion of higher excitations leads to a systematic improvement in accuracy, but at the expense of a substantial increase in computational cost. The CCSDT method69,70 scales as N8 and the inclusion of quadruples excitation (CCSDTQ)71 advances the computational scaling to N10. The coupled-cluster method shares an intimate relationship with the many-body perturbation theory (MBPT).72 Suitable lower-order iterations of coupled-cluster equations recover the various orders of MBPT. For example, the lowest-order approximation to the coupled-cluster T2 amplitudes leads to the standard MBPT(2) method. The single-reference coupled-cluster method includes dynamic correlations in a systematic way. However, it fails to account for the non-dynamic correlation, which arises due to the quasidegenerate states that prevail in radicals, bond stretching, and molecular excited states. Consequently, the single-reference coupled-cluster methods, especially in the CCSD approximation, perform poorly in the above-mentioned cases. A multireference coupled-cluster (MRCC) method41−51 addresses both dynamic and nondynamic correlations in a systematic way. However, it has the active space dependency problem, as discussed previously. Along with the MRCC methods, the EOM-CC method is known for its balanced treatment of both dynamic and nondynamic correlations. In the EOM-CC method, the final states are obtained by diagonalizing the similarity transformed Hamiltonian

(1) The radical wave function has a more-complicated spin structure than that in the closed-shell molecules, in which all electrons are paired. (2) Two or more configurations make dominant contributions to the reference wave function in the radicals. The multireference perturbation theory (MRPT) method,35−38 the multireference configuration interaction (MRCI) method,39,40 and the multireference coupled-cluster (MRCC) method41−51 can avoid the above-mentioned problems. However, the results are strongly dependent on the choice of active space, which requires experience and expertise. Subsequently, these calculations cannot be performed with a mere “push of a button”. On the other hand, the equation of motion−coupled-cluster (EOM-CC) method52−55 incorporates a balanced description of the dynamic and non-dynamic correlation and presents a black box approach for the accurate calculation of energy,53−58 structure,59−61 and properties62 of open-shell molecules and molecular excited states. The EOMIP-CCSD method has been successfully used63−67 to investigate the structure and properties of problematic doublet radicals. However, despite having otherwise favorable characteristics, the EOMIP−CCSD method still has the problem of N6 scaling. Stanton and Gauss68 proposed an N5-scaled size-extensive modification of the standard EOMIPCCSD method, by approximating the effective Hamiltonian based on perturbative orders. They have coined the term EOMIP-CCSD(2) for this black box method, and they have shown favorable numerical results for the formyl radical in the DZP basis set. One molecule is too inadequate to make a benchmark and the basis set used was too small to make a definitive conclusion. However, their results were very promising and if the trend generally holds, the EOMIP-CCSD(2) method can become the method of choice for the theoretical treatment of doublet radicals. The objective of this paper is to perform a benchmark EOMIP-CCSD(2) study on the geometry and infrared (IR) spectroscopic properties for a variety of doublet radicals and compare the performance relative to experiment, with those obtained by standard EOM-IP-CCSD, singlereference MBPT(2), and CCSD methods. The paper is organized as follows. Section 2 gives a brief discussion on the theory of the EOMIP-CCSD(2) method and computational details of the calculations. The trends in the numerical results for the geometry and IR frequency of a set doublet radicals are discussed in section 3. Section 4 contains the conclusions.

H̅ = e−T HeT = (HeT )c

The subscript “c” in the above equation represents the connectedness of T with H. Since eT is not unitary, H̅ is not Hermitian. Therefore, the final states are represented by a biorthogonal set of bra and ket vectors, which are parametrized by left and right eigen vectors of H̅ , denoted by L and R, respectively.

2. THEORY AND COMPUTATIONAL DETAILS The nonvariational coupled-cluster method generates the correlated wave function from a single Slater determinant reference state by the action of an exponential operator. |ψ ⟩ = eT |ϕ0⟩

(1)

∑ tia{aa†ai} ia

T2̂ = T3̂ =

1 4

∑ tijab{aa†ab†ajai}

1 6

∑

(4)

|Ψ⟩ = eT R |ϕ0⟩

(5)

R≡

∑ rii + i

ijab

ijkabc

⟨Ψ̃| = ⟨ϕ0|Le−T

For ionization problems, ⟨Ψ̃| and |Ψ⟩ do not correspond to particle conserving operators, but rather involve the net creation (L) and annihilation (R) of one electron. L and R can be expressed in second quantization notation as follows: 1 L ≡ ∑ l ii† + ∑ laiji†aj† 2 ija i (6)

|ϕ0⟩ is generally, but not necessarily, the Hartree−Fock determinant and T = T1 + T2 + T3 + ... Tn, where T1̂ =

(3)

1 2

∑ rijaa†ij ija

(7)

Diagonalization of H̅ in the (N − 1) electron space gives the singly ionized states of an N-electron state and the theory is called the EOMIP-CC method. It is equivalent to the (0,1) sector of the

abc † † † tijk {aa ab ac ak ajai}

(2) 4314



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The diagonalization of H[2] in a space spanned by a subset of (N − 1) determinants leads to the final ionized states of N-electron molecular problem, which can be performed by slight modification of any standard EOMIP-CCSD code. The explicit derivation of the expressions has been presented in ref 68. The EOMIP-CCSD(2)method is naturally spin-adapted and equipped to deal with multireference situations, and, therefore, it is free from the problems that are associated with the standard singlereference treatment for doublet radicals. The truncation of the effective Hamiltonian, based on perturbation order, leads to an obvious loss in accuracy. However, it gains significant computational simplifications, which lead to advantages in terms of both CPU timing and storage. The computational requirements of the standard EOMIP-CCSD method are dominated by the N-electron reference state CCSD calculation. Approximating the reference state using the MBPT(1) wave function reduces the computational scaling of the reference state calculation to N5, as opposed to the iterative N6 scaling of the CCSD reference-state calculation. Now, in IP calculation, the diagonalization (using Davidson’s iterative method) of the H̅ scales as N5 and the superficially N6 scaling intermediates can also be computed in an iterative N5 scaling algorithm, by calculating them on the fly.83 Thus, overall, the EOMIP−CCSD(2) method scales as N5, which is vastly more economical84,85 than the standard EOMIP-CCSD method. The terms containing the (ab|cd)-type integral present the most computationally demanding part of the coupled-cluster iterations. The large file size of the four particle integrals often becomes the limiting factor for storage and memory. It also slows the overall speed of the calculation, by creating input/output (I/ O) bottlenecks, even for small molecules. Now, (ab|cd) integrals only contribute to the reference-state CCSD calculation and, consequently, remain totally absent in the EOM part for the ionization problem. Thus, approximating the reference-state wave function at MBPT(1) leads to a significant savings in terms of disk space, since it does not require the (ab|cd) integrals. The favorable N5 scaling and reduced storage requirements86 make the EOMIP-CCSD(2) method applicable to the systems of considerably large dimension, where the use of the normal EOMIP-CCSD method is not possible. T1 diagnosis values presented in the paper are calculated using Gaussian 09.87 All the other results presented in the paper are calculated using CFOUR.88 All of the electrons are used in the correlation calculations. Dunning’s correlation-consistent augcc-pVXZ (X = D, T, Q) basis sets89 are used in the calculations. Equilibrium geometries (re) without any vibrational averaging and harmonic vibrational frequencies are used for comparison with the experiments.

Fock space multireference coupled-cluster (FSMRCC) method for the principal ionizations.73 FSMRCC has been successfully implemented for spectra,47,49,51,74−76 properties,77 and transition moments.78,79 The energy, using the EOMIP-CC method, can be written in the following illustrative functional form: −T T E = ⟨ϕ0|LHR ̅ |ϕ0⟩ = ⟨ϕ0|Le He R |ϕ0⟩

(8)

The EOMIP-CC method is commonly used in singles and doubles approximation (EOMIP-CCSD). It has the same N6 scaling as that of the single-reference CCSD method and similar storage requirements, which prohibit its applications beyond the first-row atoms, in a moderate basis set. Thus, it is highly desirable to develop methods, similar in spirit to the standard EOMIP-CCSD but with lower computational scaling and smaller storage requirements. Nooijen and Snijders80 were the first to propose a simplification81,82 to the standard EOM-IP-CCSD method, by approximating the full CCSD similarity transformed Hamiltonian as [1]

H̅ ≡ H̅ NS = e−T He−T

[1]

(9)

H̅ NS is complete up to the first order in correlation and contains selective contributions from higher-order terms. Diagonalization of H̅ NS in the (N − 1) electron space leads to the loss of size extensivity in energy. Nooijen and Snijders had eliminated the problem by diagonalizing matrix elements of a modified operator A, in place of H̅ NS, where Aμν = ⟨ϕ0|Ωμ[H̅ NS,Ω†ν]|ϕ0⟩

(10)

Ω† = {ai; aia†a aj}, and it represents the elements of the set of creation operator string that maps the |ϕ0⟩ into the determinants spanning the diagonalization space. The commutator in the equation accounts for all of the missing second-order contributions to H̅ NS, whereas third- or higher-order terms in A do not contribute to the truncated diagonalization problem. This ensures the size extensivity in energy. However, the method does not provide a clear definition for the total energy and, thus, becomes unsuitable for studying properties of the final state. Stanton and Gauss provided an alternative approach for approximating the conventional EOM-CCSD method.68 They have expanded the effective Hamiltonian in a perturbation series: H̅ = (HeT )c = H̅ [1] + H̅ [2] + H̅ [3] + ... H̅ [n]

(11)

The bracketed superscript in the above equation represents the order in perturbation and subscript c represents the connectedness of T with H. This leads to a set of hierarchical approximations to the full H̅ , and the diagonal representation of the modified effective Hamiltonian offers a set of hierarchical approximations to the corresponding EOM-CC final states, known as EOMCCSD(n). At large values of n, H̅ [n] converges to the full H̅ and, consequently, EOMCCSD(n) converges to the standard EOM-CCSD. The approximate similarity transformed Hamiltonian, truncated at the nth order, includes only the terms up to the order n in perturbation, which ensures the size extensivity of the final energy for all values of n. Truncation at n = 2, leads to EOMCCSD(2), with a MBPT(1) ground-state reference wave function and a MBPT(2) ground-state energy. The EOMCCSD(2) method provides an energy difference value that is identical to that of the Nooijen and Snijders’s method and has the additional advantage of a clearly defined final energy.

3. RESULTS AND DISCUSSIONS To compare the timing of EOMIP-CCSD(2) with standard EOMIP-CCSD, we have calculated the ionization potential of water clusters((H2O)n, where n = 1−8) in cc-pVDZ basis set. The wall timings are presented in Table 1. The EOMIPCCSD(2) method is found to be computationally significantly less expensive than the EOMIP-CCSD method. To benchmark the reliability of the EOMIP-CCSD(2) method, we have calculated the geometry and vibrational frequencies of NO2, NO3, trans ONOO, NO, CN, F2+, CO+, O2+, and N2+ in aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis. The above-mentioned radicals present a significant challenge for conventional ab initio methods. In all of the cases, the results are compared with those obtained by standard 4315



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Table 1. Wall Timings for the EOMIP-CCSD(2) and EOMIPCCSD Methoda,b in the cc-pVDZ Basis Set

Table 3. Geometry and Harmonic Vibrational Frequency of Nitrogen Dioxide (NO2)

Wall Timing (s) number of H2O units

EOMIP-CCSD

EOMIP-CCSD(2)

1 2 3 4 5 6 7 8

1.42 5.30 20.22 83.64 236.00 550.86 1385.21 3964.00

1.40 2.30 4.77 14.63 39.86 81.69 223.21 700.91

EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2 EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2

a

All the calculations were performed using an i7 desktop with 3.40 GHz CPU speed and 16 GB of RAM. Calculations were performed using single core. bCalculations were performed assuming C1 symmetry.

EOMIP-CCSD, UMP2, ROMP2, UCCSD, and ROCCSD methods, as well as with available experimental data. To test the suitability of the EOMIP-CCSD(2) method for larger radicals, the geometry and IR frequencies have been calculated for the water dimer cation, the formate radical, the uracil cation, the acetylene dimer cation, and the benzene dimer cation as test cases, and the results are compared with those obtained using the standard EOMIP-CCSD method. A. Nitrogen Dioxide (NO2). Nitrogen dioxide (NO2) is a very important molecule in atmospheric chemistry; consequently, it is subjected to many computational investigations.63,90−92 The triplet instabilities and second-order Jahn−Teller (SOJT) effect in NO2, lead to the mixing of 2A1 and 2B2 states, which makes the description of vibrational modesespecially the asymmetric stretching problematic. The T1 diagnosis value of 0.026 (Table 2)

EOMIP-CCSD2 EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2

a

T1 diagnosis

NO CN F2 + CO+ O2+ N2 + NO2 NO3 ONOO

0.023 0.050 0.013 0.046 0.014 0.022 0.026 0.035 0.034

bond angle (θ)

ω1 (cm−1)

aug-cc-pVDZ Basis Set 1.2000 134.2 756 1.201 133.3 772 1.299 124.51 577 1.197 134.7 767 1.284 125.0 671 1.216 132.6 753 aug-cc-pVTZ Basis Set 1.186 134.9 769 1.186 133.7 795 1.283 124.9 139i 1.181 135.2 785 1.270 125.4 609 1.201 133.2 763 aug-cc-pVQZ Basis Set 1.185 135.7 773 1.185 133.4 802 1.282 124.8 169i 1.180 135.0 790 1.269 125.23 614 1.200 133.0 767 Experimental Results 1.194a 133.9a 750b

ω2 (cm−1)

ω3 (cm−1)

1370 1413 955 1412 1138 1317

1754 1719 908i 1721 1629 1844

1384 1425 596 1438 1273 1329

1784 1745 1190 1762 1071i 1844

1391 1450 603 1443 1279 1331

1782 1744 1199 1762 1073i 1840

1325b

1634b

b

Values taken from ref 93. Values taken from ref 94.

well-described in the EOMIP-CCSD method. However, it shows more error than the EOMIP-CCSD(2) method for the bending mode (|Δωe| = 22 cm−1), as well as symmetric stretching mode (|Δωe| = 88 cm−1). The UCCSD method performs very poorly for both geometry and vibrational frequencies; it especially underestimates the asymmetric stretching mode by a large value of 726 cm−1. This is due to instabilities associated with the unrestricted reference wave function, which is indicated by a negative eigen value of the orbital rotation Hessian. The same reason leads to the disastrous performance of the UMP2 method. The performance of the ROCCSD method is similar to that of the EOMIP-CCSD method, with regard to geometry and harmonic vibrational frequencies. The ROMP2 method, on the other hand, shows more error than the EOM and ROCCSD methods for bond length (|Δre| = 0.022 Å) and bond angle (|Δangle| = 1.3°), but performs significantly better than both the UCCSD and UMP2 methods. The ROMP2 method performs surprisingly well for bending and symmetric stretching modes of vibration. However, it significantly overestimates the asymmetric stretching mode (|Δωe| = 210 cm−1). Upon moving from the aug-cc-pVDZ basis set to the aug-ccpVTZ basis set, the bond length shrinks and the bond angles are stretched for all of the theoretical methods. The EOMIPCCSD(2) method continues to give a similar performance to that of the EOMIP-CCSD method, with regard to geometry, but gives better agreement with the experiment in the case of harmonic vibration frequencies. However, both methods overestimate the experimental asymmetric stretching frequency. The UCCSD method continues to give poor performance, with regard to geometry and harmonic vibrational frequencies. The imaginary value of the bending mode indicates that the optimized geometry in the UCCSD method is not actually the minimum on the potential energy surface, but rather is a first-order saddle point. The ROCCSD method avoids the disastrous failure of the

Table 2. T1 Diagnosis Value of the Doublet Radicals molecule

bond length (Å)

method

indicates significant multireference character of the molecule. Table 3 presents the geometry and the vibrational frequencies for NO2. In the aug-cc-pVDZ basis set, the EOMIP-CCSD(2) method shows excellent agreement with the experiment for both bond lengths (|Δre| = 0.006 Å) and bond angles (|Δangle| = 0.3°). The method also provides very good agreement with the experiment93,94 for the bending mode of vibration (|Δωe| = 6 cm−1). However, it overestimates both symmetric stretching modes ((|Δωe| = 45 cm−1) and asymmetric stretching modes (|Δωe| = 120 cm−1), which are consistent with the previous theoretical reports.61,88 The performance of the EOMIP-CCSD method is similar to that of the EOMIP-CCSD(2) method, for both bond lengths (|Δre| = 0.007 Å) and bond angles (|Δangle| = 0.6°). The asymmetric stretching (|Δωe| = 85 cm−1) mode is comparatively 4316



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Table 4. Geometry and Harmonic Vibrational Frequency of Nitrogen Trioxide (NO3) Vibrational Frequency (cm−1)

Bond Length (Å) method

L1

L2

ω1 (asym bend)

EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2

1.240 1.235 1.261 1.261 1.284 1.267

1.240 1.235 1.197 1.198 1.204 1.220

163 310 528 7792i 682 521


1.228 1.221 1.247 1.247 1.270 1.255

1.228 1.221 1.182 1.183 1.191 1.206

66 305 552i 538 701 526


1.227 1.219 1.246 1.246 1.269 1.254

1.227 1.219 1.182 1.182 1.190 1.205

95 314 547 442 703 526

1.240a

250b

1.240a a

b

ω2 (asym bend)

ω3 (umbrella)

augcc-pVDZ Basis Set 163 783 310 813 630 810 520 785 702 667 655 833 aug-cc-pVTZ Basis Set 66 800 305 836 857 826 619 807 908i 615 663 842 aug-cc-pVQZ Basis Set 95 801 314 838 7917 779 623 809 910i 612 664 843 Experimental Results 250b 762c

ω4 (sym stretch)

ω5 (asym stretch)

ω6 (asym stretch)

1130 1149 1129 1116 1058 1087

1198 1183 1102 1647 791i 1642

1198 1183 1667 277833 1650 2097

1140 1170 1174 1139 1081 1097

1176 1191 581 998 716 1643

1176 1191 1700 1778 1648 2123

1142 1174 1152 1143 1085 1099

1175 1192 1693 1005 718 1644

1175 1192 27249i 1682 1650 2126

1060c

1480c

1480c

c

Values taken from ref 98. Values taken from ref 95. Values taken from ref 96.

coupled-cluster (FSMRCC) calculations by Kaldor,103 and MRCI calculations by Morkuma and Eisfield,104 both resulted in a D3h ground-state geometry for the NO3 radical. The EOMIP-CCSD(2) and EOMIP-CCSD methods predict a D3h ground-state geometry. In the aug-cc-pVDZ basis set, the EOMIP-CCSD(2) method shows better agreement with the experimental bond length than the EOMIP-CCSD method. Both methods slightly underestimate the bond length in the aug-ccpVTZ and aug-cc-pVQZ basis sets. However, the EOMIPCCSD(2) method continues to give better agreement with the experiment than the EOMIP-CCSD method. The UCCSD and ROCCSD methods predict a C2v minimum energy structure (2L1S) for the ground state and both the methods predict identical geometries. In the aug-cc-pVDZ basis set, both methods overestimate the experimental bond length, in the case of the long bonds (2L) and underestimate the experimental bond length in the case of the short bond (1S). The two long bonds shrink in the aug-cc-pVTZ and aug-cc-pVQZ basis sets, bringing them closer to the experimental value. However, the use of a large basis set also shrinks the shorter bond, taking it further away from the experimental value. The UMP2 and ROMP2 methods follow the trend of their coupled-cluster analogues, only with a larger error bar. The EOMIP-CCSD(2) method performs reasonably well for the IR frequencies of NO3. It predicts the umbrella (ω3) and the symmetric stretching (ω4) mode of vibrations with high accuracy. However, it underestimates the low-frequency asymmetric bending (ω1 and ω2) and high-frequency asymmetric stretching (ω5 and ω6) modes (see Table 4). The umbrella and symmetric stretching modes shift to higher values, when using larger basis sets; whereas, all of the other modes of vibration shrink to lower values. While the EOMIP-CCSD method shows significant improvement over the EOMIPCCSD(2) method, for the asymmetric bending mode, it reduces

UCCSD method, but gives inferior performance, compared to the EOM methods. The UMP2 method suffers from the spin contamination problem, similar to that of the UCCSD method; consequently, the results show a large deviation from the experimental bond length and bond angle. The asymmetric stretching mode in the UMP2 method shows an imaginary value, indicating that the geometry is a first-order saddle point. The ROMP2 method gives remarkable agreement with the experimental values with regard to geometry, as well as bending and stretching modes of vibration, but it overestimates the asymmetric stretching mode of vibration, and shows greater error (|Δωe| = 204 cm−1) than the EOM-based methods. The same trend persists in the aug-cc-pVQZ method, where the geometries and IR frequencies for all of the methods show very little deviation from their corresponding values in the augcc-pVTZ basis set. B. Nitrogen Trioxide (NO3). The ground-state geometry of the nitrogen trioxide (NO3) radical has been an issue of longstanding debate. Experimental studies95−99 have provided evidence in favor of D3h geometry, whereas various theoretical studies have predicted different minimum energy structures for the molecule. Three structures have been found to be energetically most favorable: (a) a highly symmetric D3h structure, (b) a C2v structure with two long bonds and one short bond (2L1S), and (c) C2v structure with one long bond and two short bonds (1L2S). Initial MCSCF studies100 have predicted a Y-shaped structure for the NO3 radical. In 1992, Bartlett and co-workers101 have reported the C2v structure to be 2.5 kcal/mol lower in energy than the more-symmetric D3h structure in the B-CCD level of theory. Crawford and Stanton102 latter revised this ordering, by placing the D3h structure 0.5 kcal/ mol below the C2v structure in the B-CCD(T) method. The T1 diagnosis value of 0.035 (Table 2) indicates significant multireference character for the species. Fock space multireference 4317



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Table 5. Geometry and Harmonic Vibrational Frequency of Trans Nitro Peroxide (OONO) Bond Length (Å)

Vibrational Frequency (cm−1)

Bond Angle (deg)

method

O−O

N−O

O−N


1.285 1.298 1.216 1.352 1.259 1.328

1.557 1.598 3.316 1.605 3.146 1.762

1.157 1.149 1.159 1.143 1.141 1.137


1.275 1.287 1.196 1.281 1.243 1.229

1.537 1.556 3.224 1.582 3.034 1.183

1.144 1.136 1.142 1.136 1.135 1.136


1.274 1.287 1.196 1.337 1.226 1.227

1.540 1.557 3.332 1.564 3.229 1.849

1.143 1.135 1.141 1.129 1.135 1.136

N−O−O

O−N−O

ω1

aug-cc-pVDZ Basis Set 109.5 108.1 193 106.6 108.2 150 88.3 166.1 61i 110.1 108.3 192 73.4 171.0 8 110.9 108.7 170 aug-cc-pVTZ Basis Set 110.1 108.2 199 107.4 108.3 158 79.3 169.2 102i 107.8 108.2 181 78.2 186.1 50 107.8 110.2 1916i aug-cc-pVQZ Basis Setb 110.2 108.3 198 107.5 108.3 157 79.6 169.6 17 111.1 108.6 200 79.1 169.1 9i 108.2 110.2 117 Experimental Results

ω2

ω3

ω4

ω5

ω6

361 351 33 260 58 226

452 421 36 368 65 310

881 848 36 701 170 675

1287 1207 1581 929 1219 913

1800 1902 1946 1900 3658 1980

366 368 79 280 40 113

468 450 181 338 79 105

901 870 277 723 441 522

1307 1251 1646 1170 1329 1055

1815 1925 1957 1867 3139 1805

364 368 34 272 41 197

465 451 43 393 50 202

897 870 53 731 63 700

1307 1250 1673 989 1428 1336

1811 1919 2000 1912 3169 1924 1840a

a

b

Value taken from ref 107. Diffused g functions were excluded from the basis to keep the calculations computationally viable.

C. Trans Nitro Peroxide (OONO). The correlation of experiments and theory for the trans nitro peroxide has been a matter of long-standing debate.105,106 Bhatia and Hall109 have suggested a planar trans structure from IR spectroscopic investigation. However, later ab initio studies,105,108,109 using different levels of theories, have shown contradictory results. The radical shows considerable multireference character (T1 diagnosis value = 0.034), making it a challenging case for standard ab initio methods. Geometry and IR frequencies of trans ONOO, computed at different levels of theory, are presented in Table 5. The EOMIP-CCSD(2) method shows an O−N bond length of 1.157 Å in the aug-cc-pVDZ basis set. The O−O bond length (1.285 Å) shows considerable elongation from the bare molecular oxygen bond length of 1.207 Å and, consequently, the N−O bond length (1.157 Å) decreases from the free nitric oxide bond length value of 1.160 Å. It indicates an electron transfer from the antibonding orbital of nitric oxide to the antibonding orbital of oxygen, leading to shrinkage of the former and stretching of the latter. In the EOMIP-CCSD(2) method, the IR frequency corresponding to the N−O bond stretching mode (ω6) is underestimated by 40 cm−1. The EOMIP-CCSD method predicts longer O−O and O−N bond lengths, compared to the EOMIP-CCSD(2) method. On the other hand, the N−O bond shrinks, from 1.157 Å to 1.149 Å, upon moving from the EOMIP-CCSD(2) method to the EOMIP-CCSD method, which leads to overestimation of experimental frequency (62 cm−1) in the latter. The ROCCSD and ROMP2 methods follow the same trend as that of the EOMIP-CCSD method. However, the UCCSD method predicts a weakly bound T-shaped geometry (Figure 1) with an elongated O−N bond length of 3.316 Å. The O−O and the N−O bond lengths are almost same as that of their free molecular form. The structure shows an imaginary frequency, which indicates that the geometry is a

the accuracy of the umbrella and symmetric stretching modes. Both EOMIP-CCSD(2) and EOMIP-CCSD methods, however, significantly underestimate the asymmetric stretching mode. Here, it should be noted that the experimental assignments of the asymmetric stretching at 1492 cm−1 is not unambiguous.64 Detailed studies are required to make a concluding statement about these problematic modes, which are beyond the scope of the present study. The single-reference coupled-cluster methods show large errors for the harmonic vibrational frequencies. The UCCSD method, in aug-cc-pVDZ basis, overestimates the two asymmetric bending modes and one of the asymmetric stretching modes, while it underestimates the other asymmetric stretching mode. These trends continue in the aug-cc-pVTZ basis, where the ω1 shows an imaginary frequency, indicating that the geometry is a first-order saddle point. The UCCSD method in the aug-cc-pVQZ basis set shows a unphysical value for one of the asymmetric stretching modes (ω2) and one of the asymmetric bending modes (ω6) of vibration. The instability in the UHF reference wave function, indicated by the negative eigen value of the orbital rotation Hessian, leads to the catastrophic failure of the UCCSD method in the present case. The UMP2 method follows the same trend as that of its coupled-cluster analogue and shows one imaginary frequency in all of the basis sets. The ROCCSD method shows an imaginary frequency for the asymmetric bending mode (ω1) in the aug-cc-pVDZ basis set, indicating that the geometry is a first-order saddle point. It also gives an unphysical value for the asymmetric stretching mode (ω6). Although these spurious results vanish at higher augcc-pVTZ and aug-cc-pVQZ basis sets, the calculated IR frequencies still show large deviations from the experimental results. The ROMP2 method closely follows the trend of ROCCSD method, with the difference being that the former does not show any unphysical or imaginary frequencies in any of the basis sets used. 4318



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Table 6. Geometry and Harmonic Vibrational Frequency of Nitric Oxide method EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2

Figure 1. EOMIP-CCSD(2) and UCCSD optimized structure of trans nitro peroxide (ONOO) in the aug-cc-pVQZ basis set.

saddle point of order one. The IR frequency corresponding to the ω6 mode shows a value of 1946 cm−1, which is 106 cm−1 higher than the experimental value. The UMP2 method also shows a T-shaped structure, but with a larger O−O bond, and a shorter O−N bond, as well as a shorter N−O bond. The UMP2 method predicts an N−O bond stretching frequency of 3658 cm−1, which is almost double the experimental frequency of 1840 cm−1. The UHF instability, which is indicated by the negative value of the orbital rotation Hessian, is responsible for the unphysical behavior of the N−O stretching in the UMP2 method. The EOM methods, as well as the UCCSD method, lead to shrinkage of all of the bonds and an upward shifting of the IR frequencies in the aug-cc-pVTZ basis set. The ROCCSD and ROMP2 methods show similar trends for the bond length, but the IR frequency corresponding to the ω6 mode undergoes a downward shift in both methods. The UMP2 method also shows a large downward shift of the ω6 mode, but still overestimates the experimental frequency by more than a thousand wavenumbers. The results in the EOMIP-CCSD(2) and EOMCCSD methods in the aug-cc-pVQZ basis set show very small deviations from those in the aug-cc-pVTZ basis set. The EOMIP-CCSD(2) method gives the best agreement (|Δωe| = 29 cm−1) with the experimental frequency, among all the methods used. However, the UCCSD and UMP2 methods result in a longer O−N bond in the aug-cc-pVQZ basis set. It is interesting to note that both the N−O bond length and the N−O stretching frequency in the UCCSD and UMP2 methods are very similar to that in the free nitric oxide. It indicates that trans ONOO is a very weakly bound complex, contrary to the stable structure predicted by the EOM methods. In the aug-cc-pVQZ basis set, the ROCCSD method shows shrinkage of the N−O bond, which is reflected in the upward shift of the ω6 mode. The ROMP2 method also results in an upward shift of the N−O stretching frequencies in the aug-ccpVQZ basis set, but the N−O bond length remains unchanged from that in the aug-cc-pVTZ basis set. D. The Diatomics. We have tested the performance of EOMIP-CCSD(2) method for diatomic molecules such as NO, CO+, CN, F2+, O2+, and N2+. These diatomic molecules suffer from the notorious symmetry breaking problem and often represent a classic challenge for standard single-reference theories. Nitric oxide acts as a catalyst for the ozone depletion reactions and plays a key role in stratospheric ozone chemistry. Table 6 contains the computed bond length and IR frequency of nitric oxide. In the aug-cc-pVDZ basis set, all the methods overestimate the N−O bond length, compared to the experimental value. The IR frequency in all of the methods shows reasonable agreement with the experiment, except in the UMP2 method, where the experimental frequency is overestimated by 1742 cm−1. The bond length shrinks and, consequently, the IR frequency shifts to a higher value in both the aug-cc-pVTZ and aug-cc-pVQZ basis sets. The EOMIP-CCSD(2) method underestimates the experimental bond length, but gives better agreement compared to the ROCCSD and UCCSD methods. For IR frequencies also, it gives a value (|Δωe| = 118 cm−1) comparable to the

EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2 EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2

a

bond length (Å) aug-cc-pVDZ Basis Set 1.160 1.164 1.160 1.159 1.142 1.170 aug-cc-pVTZ Basis Set 1.146 1.150 1.142 1.142 1.135 1.154 aug-cc-pVQZ Basis Set 1.145 1.150 1.141 1.141 1.341 1.154 Experimental Results 1.150a

Values taken from Huber and Herzberg.

frequency, ω (cm−1) 1981 1957 1941 1958 3646 1897 2014 1996 2000 2000 3179 1920 2022 2005 2012 2005 3166 1922 1904a

110

EOMIP-CCSD (|Δωe| = 101 cm−1), UCCSD (|Δωe| = 112 cm−1), and ROCCSD (|Δωe| = 101 cm−1) values, in the aug-cc-pVQZ basis set. The UMP2 method performs very poorly, with regard to bond length (|Δre| = 0.019 Å) and IR frequency (|Δωe| = 1262 cm−1). The ROMP2 method, on the other hand, shows surprisingly close agreement with the experiment,110 with regard to bond length and IR frequency. N2+ shows a T1 diagnostic value of 0.022 (Table 2); therefore, single-reference methods can treat it with reasonable accuracy. Table 7 shows that, in the aug-cc-pVDZ basis set, the EOMIPCCSD method shows the best agreement with the experiment, with regard to bond length, but overestimates the experimental IR frequency by more than a hundred wavenumbers. The EOMIP-CCSD(2) method, on the other hand, overestimates the bond length but reproduces the experimental110 IR frequency (|Δωe| = 5 cm−1) with high accuracy. The UHF- and ROHFbased single-reference coupled-cluster methods give similar accuracy as that of the EOMIP-CCSD(2) method for bond length, but lead to greater error in IR frequency. Calculations in the aug-cc-pVTZ basis set result in shrinkage of the bond length and an upward shift of vibrational frequency. The results in the aug-ccpVQZ basis set show negligible deviation from that in the aug-ccpVTZ basis. The EOMIP-CCSD(2) method shows the best agreement with experiment for both bond length (|Δre| = 0.004 Å) and IR frequency (|Δωe| = 42 cm−1). The UCCSD and ROCCSD methods underestimate the bond length and consequently overestimate the IR frequency, by 94 and 101 cm−1, respectively. The MBPT(2) methods overestimate the bond length and significantly underestimate the IR frequency. N2+ shows two significant exceptions to the trend shown by the previously discussed molecules. First, the UMP2 method performs better than the ROMP2 method; second, the EOMIP-CCSD method performs poorly for both bond length (|Δre| = 0.010 Å) and IR frequency (|Δωe| = 152 cm−1). 4319



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compared to the EOMIP-CCSD and single-reference coupledcluster methods, but performs significantly better than both the UMP2 method (|Δre| = 0.049 Å) and the ROMP2 method (|Δre| = 0.062 Å). However, the EOMIP-CCSD(2) method gives excellent agreement with the experimental value for IR frequency. The aug-cc-pVDZ basis set, however, is inadequate to rely upon. Calculations in the aug-cc-pVTZ and aug-cc-pVQZ basis sets lead to the contraction of the bond length and increment of the IR frequency, in all of the methods used. In the aug-cc-pVQZ basis set, the EOMIP-CCSD(2) method shows the best agreement with the experiment110 for both bond length (|Δre| = 0.004 Å) and harmonic vibrational frequency (|Δωe| = 37 cm−1). The EOMIP-CCSD method gives comparable performance for bond length (|Δre| = 0.009 Å), but performs poorly for IR frequency (|Δωe| = 117 cm−1). The single-reference coupledcluster methods also show inferior results for both bond length (|Δre| = 0.013 and 0.012 Å for the UCCSD and ROCCSD methods, respectively) and harmonic vibrational frequency (|Δωe| = 157 and 162 cm−1 for the UCCSD and ROCCSD methods, respectively). The UMP2 method performs very poorly with regard to bond length (|Δre| = 0.030 Å) and IR frequency (|Δωe| = 362 cm−1). However, the predicted values are better than those in the ROMP2 method (|Δre| = 0.041 Å and |Δωe| = 476 cm−1). O2+ follows the unique trend shown by MBPT(2) and the EOMIP-CCSD method in N2+, as discussed in the previous paragraph. The T1 diagnosis value (0.050) indicates significant multireference character of the CN radical. Table 9 lists the computed

Table 7. Geometry and Harmonic Vibrational Frequency of N2+ method EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2 EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2 EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2

a



frequency, ω (cm−1) 2212 2316 2245 2253 2078 2008 2250 2359 2299 2306 2124 2055 2249 2359 2301 2308 2123 2054 2207a

110

O2+ shows a very small T1 diagnosis value of 0.014, which indicates that a single determinant reference will be sufficient for the accurate description of the wave function. Table 8 shows that, in the aug-cc-pVDZ basis set, the EOMIP-CCSD(2) method exhibits greater error for the bond length (|Δre| = 0.011 Å),

Table 9. Geometry and Harmonic Vibrational Frequency of CN method

Table 8. Geometry and Harmonic Vibrational Frequency of O2+ method EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2 EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2 EOMIP-CCSD(2) EOMIP-CC UCCSD ROCCSD UMP2 ROMP2

a



frequency, ω (cm−1) 1908 1981 2016 2022 1457 1326


1931 2009 2049 2054 1526 1410

EOM-IP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2

1942 2022 2062 2067 1543 1429

a



frequency, ω (cm−1) 1805 1877 2117 2104 2843 1753 2137 2178 2187 2167 2916 1849 2133 2174 2188 2164 2923 1847 2069a

110

bond lengths and IR frequencies of the CN radical. In the aug-ccpVDZ basis set, however, both EOM methods give disastrous performance for bond length and IR frequency.

1905a

Values taken from Huber and Herzberg.110 4320



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the ROMP2 method underestimates the frequency by 68 cm−1. In the aug-cc-pVTZ and aug-cc-pVQZ basis sets, both the EOM and single-reference coupled-cluster methods underestimate the bond length and overestimate the frequency. The UCCSD and ROCCSD methods, in the aug-cc-pVQZ basis set, give significant error in bond length (|Δre| = 0.041 and 0.040 Å, respectively), as well as in frequency (|Δωe| = 166 and 170 cm−1, respectively). The EOMIP-CCSD(2) method gives better performance than the single-reference coupled-cluster method, although it continues to underestimate the bond length (|Δre| = 0.027 Å), and overestimates the harmonic vibrational frequency (|Δωe| = 105 cm−1). The EOM-CCSD method gives a performance similar to that of the EOMIPCCSD(2) method, in both the aug-cc-pVTZ and aug-ccpVQZ basis sets. The UHF- and ROHF-based MBPT(2) methods overestimate the bond length and underestimate the IR frequency. The UMP2 shows a surprisingly accurate bond length of 1.341 Å and leads to an IR frequency that has accuracy comparable to that of the single-reference coupledcluster method. However, the ROMP2 method performs very poorly for both bond length (|Δre| = 0.047 Å) and IR frequency (|Δωe| = 285 cm−1) . CO+ is isoelectronic with N2+, but shows significant multireference character (T1 value = 0.046). Table 11 shows that, in

The EOMIP-CCSD(2) method overestimates the bond length by 0.076 Å and underestimates the frequency by 264 cm−1. The EOMIP-CCSD method improves the results, but still shows large error (|Δre| = 0.069 Å and (|Δωe| = 192 cm−1), compared to the experiment. This trend gets reversed in larger basis sets: the EOMIP-CCSD(2) method gives the best agreement with the experiment for both bond length (|Δre| = 0.007 Å) and vibrational frequency (|Δωe| = 64 cm−1) in the aug-cc-pVQZ basis set. The EOMIP-CCSD method slightly underestimates the bond length (|Δre| = 0.010 Å) and overestimates the frequency (|Δωe| = 105 cm−1). The performance of the singlereference coupled-cluster methods is similar to that of the EOMIP-CCSD method, with regard to both bond length and IR frequency. The spin contamination of the UMP2 wave function introduces very large errors in bond length (|Δre| = 0.050 Å) and IR frequency (|Δωe| = 854 cm−1) . The ROMP2 method shows significant improvement over the UMP2 method, but the bond length (|Δre| = 0.014 Å) and IR frequency (|Δωe| = 222 cm−1) still deviate significantly from the experiment.110 The F2+ shows a T1 diagnosis value of 0.013, which makes it a suitable candidate for single-reference treatment. Table 10 shows Table 10. Geometry and Harmonic Vibrational Frequency of F2+ method EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2 EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2 EOMIP-CCSD(2) EOMIP-CCSD UCCSD ROCCSD UMP2 ROMP2

a

bond length (Å) aug-cc-pVDZ Basis Set 1.322 1.326 1.312 1.310 1.395 1.437 aug-cc-pVTZ 1.299 1.299 1.286 1.285 1.347 1.374 aug-cc-pVQZ 1.295 1.295 1.281 1.280 1.343 1.369 Experimental Results 1.322a

Table 11. Geometry and Harmonic Vibrational Frequency of CO+

frequency, ω (cm−1)

method

1081 1065 1128 1133 1124 1005


1174 1168 1230 1234 890 781


1178 1176 1239 1243 897 788


1073a

Values taken from Huber and Herzberg.110 a

that, in the aug-cc-pVDZ basis set, the EOMIP-CCSD(2) method reproduces the experimental110 bond length and frequency with absolute accuracy. The EOMIP-CCSD method also gives comparable performance for bond length and IR frequency. However, the single-reference coupled-cluster methods underestimate the bond length and overestimate the IR frequency. The UMP2 and ROMP2 methods also overestimate the bond length by a large value (0.073 and 0.115 Å, respectively), but give IR frequencies with accuracy comparable to that of their coupled-cluster analogues. The UMP2 method overestimates the experimental frequency by 51 cm−1, whereas



frequency, ω (cm−1) 2221 2257 2248 2259 2850 2065 2282 2324 2322 2327 2881 2129 2288 2331 2330 2333 2888 2132 2214a

110

the aug-cc-pVDZ basis set, the EOM and the single-reference coupled-cluster methods overestimate the bond length. For IR frequency, the EOMIP-CCSD(2) method gives the best agreement with the experiment110 (|Δωe| = 7 cm−1), whereas the EOMIP-CCSD and single-reference coupled-cluster methods result in overestimation of the frequency. The UMP2 method underestimates the bond length (|Δre| = 0.016 Å) and overestimates the frequency (|Δωe| = 636 cm−1) by a considerable margin. The ROMP2 method shows greater error 4321



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than the UMP2 method, with regard to bond length (|Δre| = 0.023 Å), but gives a better result for harmonic vibrational frequency (|Δωe| = 149 cm−1). In the aug-cc-pVTZ basis set, the bond length decreases and the IR frequency increases for all of the methods. The EOMIP-CCSD(2) method gives the best agreement with experiment for bond length (|Δre| = 0.006 Å), as well as IR frequency (|Δωe| = 66 cm−1). The EOMIP-CCSD method, as well as the single-reference coupled-cluster method, show inferior performance than the EOMIP-CCSD(2) method in the aug-cc-pVTZ basis set. The UMP2 method severely underestimates the bond length (|Δre| = 0.027 Å) and overestimates the frequency (|Δωe| = 666 cm−1). However, the

ROMP2 method gives surprisingly good performance for the bond length (|Δre| = 0.009 Å) as well as the vibrational frequency (|Δωe| = 79 cm−1). The same trend holds in the aug-cc-pVQZ basis set, and the bond length and IR frequency show very small deviations from that determined in the aug-ccpVTZ basis set. E. Error Analysis. Tables 12 and 13 present the minimum, maximum, and average absolute deviations of the computed (in the aug-cc-pVQZ basis set) bond lengths and harmonic vibrational frequencies from the experiment, for all of the molecules investigated in the paper. Among all the methods used in this work, the EOMIP-CCSD(2) method shows the lowest average absolute deviation for bond length (|Δre| = 0.010 Å) as well as harmonic vibrational frequency (|Δωe| = 111 cm−1); whereas the UCCSD and UMP2 methods show the highest maximum absolute deviations for bond length and harmonic vibrational frequency, respectively. Figure 2 and 3 reveal that the EOMIP-CCSD(2) method gives the best performance for both bond length and IR frequency; these values are very close to (and even better than, in some cases) the EOMIP-CCSD method, despite the latter being computationally more demanding. It is followed, in order of decreasing accuracy, by the ROCCSD method, the ROMP2 method, the UCCSD method, and, lastly, the UMP2 method. F. Some More Test Cases. After gaining some confidence about the reliability of the EOMIP-CCSD(2) method for structure and IR frequency, we proceed to investigate some larger doublet radicals and dimer radical cations. The latter often show serious symmetry breaking and spin-contamination problems, as previously reported by Krylov and coworkers111,112 as test cases for estimating the accuracy of the EOMIP-based methods. We compare the structural parameters and IR frequencies calculated using the EOMIPCCSD(2) method with those obtained using the standard EOMIP-CCSD method, in different basis sets. Water Dimer Cation. Ionization introduces considerable change in the electronic structure of water dimer cation and leads to the formation of the OH···H3O+ complex, as seen from Figure 4. The structural parameters are reproduced quite well by the EOMIP-CCSD(2) method. The AAD values in both the bond length and bond angle are 0.013 Å and 0.20°, respectively.

Table 12. Comparison of the Maximum, Minimum, and Average Absolute Deviation Values of the Computed (aug-ccpVQZ Basis Set) Equilibrium Bond Lengths from the Experiment Bond Length, |Δre| (Å) method

min

max

average absolute deviation, AAD

IP-EOM-CCSD(2) EOM-IP-CCSD UCCSD ROCCSD UMP2 ROMP2

0.004 0.000 0.006 0.006 0.012 0.004

0.027 0.027 0.088 0.058 0.075 0.047

0.010 0.013 0.030 0.016 0.038 0.019

Table 13. Comparison of the Maximum, Minimum, and Average Absolute Deviation Values of the Computed (aug-ccpVQZ Basis Set) Harmonic Vibrational Frequencies from the Experiment Harmonic Vibrational Frequencies, |Δωe| (cm−1) method

min

max

average absolute deviation, AAD

IP-EOM-CCSD(2) EOM-IP-CCSD UCCSD ROCCSD UMP2 ROMP2

23 52 17 40 25 6

317 300 722 487 1337 634

111 124 233 155 482 197

Figure 2. Comparison of the maximum, minimum, and average absolute deviations of the computed (aug-cc-pVQZ basis set) bond length from the experiment. 4322



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Figure 3. Comparison of the maximum, minimum, and average absolute deviation of the computed (aug-cc-pVQZ basis set) harmonic vibrational frequency from the experiment.

Figure 4. Larger radicals and dimer cations.

insensitive to the basis set used. The geometrical parameters and harmonic vibrational frequencies of the water dimer cation are given in Tables 14 and 15, respectively. Formate Radical. The formate radical has a C2v symmetry, and its electronic wave function belongs to the B2 irreducible representation. The displacement of the nuclei along the asymmetric stretching mode leads to the coupling of the ground state with a low-lying 2A1 state (second-order Jahn− Teller interaction), which complicates its electronic structure. Therefore, an accurate description of the wave function requires a balanced treatment between the two degenerate states, so the system acts as a sensitive test case for ab initio methods.

Bond lengths shrink, upon moving from the aug-cc-pVDZ basis set to the aug-cc-pVTZ basis set, in both the EOMIPCCSD(2) and EOMIP-CCSD methods. However, the change from the aug-cc-pVTZ basis set to the aug-cc-pVQZ basis set is very small. An important thing to notice is that the errors observed in the EOMIP-CCSD(2) method, relative to those observed in the EOMIP-CCSD method are not sensitive to the basis set, as one might expect due to the different level of electron correlation included in the reference state of both methods. The IR frequencies obtained in EOMIP-CCSD(2) method also show negligible deviation from those obtained using the EOMIP-CCSD method, and the average absolute error is quite 4323



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Table 14. Geometrical Parameters of the Water Dimer Cation aug-cc-pVDZ

aug-cc-pVTZ

aug-cc-pVQZ

EOMIP-CCSD

EOMIP-CCSD(2)

error

EOMIP-CCSD

EOMIP-CCSD(2)

error

EOMIP-CCSD(2)

bond length (Å) H1−O1 H2−O1 O1−O2 H3−O2 H4−O2 AADa

0.983 1.446 2.495 0.974 0.974 0.013

0.982 1.482 2.523 0.977 0.977

0.001 0.036 0.028 0.003 0.003

0.975 1.425 2.470 0.968 0.967 0.013

0.974 1.459 2.497 0.971 0.971

0.001 −0.034 −0.027 −0.003 −0.004

0.973 1.458 2.495 0.970 0.970

bond angle (deg) H1−O1−H2 H3−O2−H4 AADb

121.18 109.67 0.20

121.39 109.59

0.11 0.08

121.8 110.3 0.20

122.08 110.13

−0.25 0.14

122.65 110.29

a

Average absolute deviation of bond length, |Δre|. bAverage absolute deviation of bond angle, |Δangle|.

Table 15. Harmonic Vibrational Frequencies of the Water Dimer Cation Harmonic Vibrational Frequencies (cm−1) aug-cc-pVDZ 1 2 3 4 5 6 7 8 9 10 11 AADa a

aug-cc-pVTZ

aug-cc-pVQZ

EOMIP-CCSD

EOMIP-CCSD(2)

error

EOMIP-CCSD

EOMIP-CCSD(2)

error

EOMIP-CCSD(2)

105 383 410 513 632 1068 1641 1733 2320 3649 3701 35

99 369 387 492 605 1006 1624 1693 2494 3650 3699

6 14 23 21 27 62 17 40 174 1 2

107 381 425 529 634 1072 1669 1766 2325 3707 3756 36

102 370 400 510 605 1013 1650 1723 2490 3696 3749

5 11 25 19 29 59 19 43 165 11 7

107 376 396 503 601 1005 1647 1712 2493 3698 3754

Average absolute deviation of harmonic vibrational frequency, |Δωe|.

Table 16. Geometrical Parameters of the Formate Radical aug-cc-pVDZ

aug-cc-pVTZ

aug-cc-pVQZ

EOMIP-CCSD

EOMIP-CCSD(2)

error

EOMIP-CCSD

EOMIP-CCSD(2)

error

EOMIP-CCSD(2)

bond length (Å) C1−O2 C1−O3 H1−C AADa

1.261 1.261 1.102 0.004

1.266 1.266 1.099

0.005 0.005 0.003

1.246 1.246 1.085 0.004

1.252 1.252 1.084

0.006 0.006 0.001

1.250 1.250 1.086

bond angle (deg) O−C1−O H1−C1−O AADb

112.40 123.80 1.19

110.80 124.59

1.6 0.79

112.45 123.77 1.16

110.90 124.55

1.55 0.78

110.92 124.54

a


set, we have performed the calculations only using the EOMIP-CCSD(2) method, and the values show very little deviation from that in the aug-cc-pVTZ basis set. The average absolute deviation in bond lengths and bond angles are 0.004 Å and 1.19°, respectively, and these values are insensitive to the basis set used.

Table 16 shows that the EOMIP-CCSD(2) method reproduces the bond lengths and bond angles, obtained using the EOMIP-CCSD method, with reasonable accuracy in both the aug-cc-pVDZ and aug-cc-pVTZ basis sets. For both methods, all the bond lengths shrink upon moving from the aug-cc-pVDZ basis set to the aug-cc-pVTZ basis set. In the aug-cc-pVQZ basis 4324



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Table 17. Harmonic Vibrational Frequencies of the Formate Radical Harmonic Vibrational Frequencies (cm−1) aug-cc-pVDZ 1 2 3 4 5 6 AADa| a

aug-cc-pVTZ

aug-cc-pVQZ

EOMIP-CCSD

EOMIP-CCSD(2)

error

EOMIP-CCSD

EOMIP-CCSD(2)

error

EOMIP-CCSD(2)

652 1021 1073 1296 1489 3152 24

634 1001 1037 1274 1467 3176

18 20 36 22 22 24

665 1052 1107 1308 1529 3164 30

643 1028 1052 1280 1498 3186

22 24 55 28 31 22

648 1024 1052 1293 1502 3188


Table 18. Geometrical Parameters of the Uracil Cation cc-pVDZ EOMIPCCSD bond length (Å) C−C(1) C−N(1) N−C(1) C−N(2) N−C(2) C−C(2) C−O(1) C−O(2) AADa bond angle (deg) C−C−N(1) C−N−C(1) N−C−N(1) C−N−C(2) N−C−C(1) C−C−C(1) AADb

EOMIPCCSD(2)

Table 19. Harmonic Vibrational Frequencies of the Uracil Cation

cc-pVTZ |error|

Harmonic Vibrational Frequencies (cm−1)

EOMIPCCSD(2)

1.372 1.358 1.390 1.431 1.344 1.432 1.277 1.192 0.011

1.384 1.348 1.397 1.429 1.341 1.412 1.305 1.195

0.012 0.010 0.007 0.002 0.003 0.020 0.028 0.003

1.367 1.335 1.384 1.415 1.328 1.394 1.298 1.188

122.75 124.33 112.23 124.73 120.38 115.57 0.42

122.36 124.80 111.55 124.77 121.13 115.37

0.390 0.470 0.680 0.040 0.750 0.20

122.42 124.62 111.73 124.36 121.49 115.36

cc-pVDZ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 AADa

Average absolute deviation of bond length, |Δre|. bAverage absolute deviation of bond angle, |Δangle|. a

The IR frequencies also show very good agreement with those obtained using the EOMIP-CCSD method, with average absolute deviations of 23 cm−1 and 30 cm−1, in the aug-cc-pVDZ and aug-ccpVTZ basis sets, respectively. The harmonic vibrational frequencies of the formate radical are given in Table 17. Uracil Cation. The investigation of ionization-induced structural change of DNA and RNA bases constitutes an important branch of radiation biology. We have investigated the geometry and vibrational frequencies of the uracil cation, which is the ionized form of the smallest of the RNA bases, so that comparison with EOMIP-CCSD method can be made in a reasonable basis set. The uracil cation has eight different bonds between the heavy atoms, as depicted in Figure 4. The bond length and bond angle in the EOMIP-CCSD(2) method, versus the EOMIP-CCSD method in the cc-pVDZ basis set, is summarized in Table 18. The EOMIP-CCSD(2) method shows reasonable agreement with the EOMIP-CCSD method, as indicated by the small average absolute deviations in bond length (0.011 Å) and bond angle (1.13°). In the EOMIP-CCSD(2) method, the

a

cc-pVTZ

EOMIP-CCSD

EOMIP-CCSD(2)

|error|

EOMIP-CCSD(2)

118 156 346 367 504 523 560 654 689 703 773 786 839 963 1011 1015 1126 1220 1245 1365 1401 1460 1522 1577 1680 1936 3266 3307 3573 3618 11

129 167 341 368 500 520 554 655 705 722 767 779 831 954 1004 1025 1115 1195 1246 1337 1376 1466 1524 1582 1656 1932 3275 3320 3547 3592

11 11 5 1 4 3 6 1 16 19 6 7 8 9 7 10 11 25 1 28 25 6 2 5 24 4 9 13 26 26

139 181 245 390 487 522 544 662 668 716 735 792 794 863 973 1045 1051 1115 1245 1281 1352 1461 1502 1623 1650 1917 3269 3335 3552 3599


bond lengths shrink, upon moving from the cc-pVDZ basis set to the cc-pVTZ basis set. Table 19 shows that both methods are in excellent agreement for the IR frequencies, exhibiting a maximum value of 28 cm−1 and an average absolute deviation of 11 cm−1. 4325



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Table 22. Geometrical Parameters of the y-Displaced Benzene Dimer Cation

Table 20. Geometrical Parameters of the Acetylene Dimer Cation cc-pVDZ EOMIPCCSD bond length (Å) H1−C1 C1−C2 C2−H2 C2−C3 C3−H3 C3−C4 C4−H4 AADa bond angle (deg) H1−C1−C2 H2−C2−C1 C1−C2−C3 H3−C3−C4 H4−C4−C3 AADb

EOMIPCCSD(2)

1.088 1.270 1.093 1.705 1.093 1.270 1.088 0.003

1.087 1.271 1.094 1.692 1.094 1.271 1.086

169.14 139.63 107.71 139.63 169.14 3.39

170.09 135.62 114.24 135.62 170.09

cc-pVTZ |error|

6-31G

EOMIPCCSD(2)

0.001 0.001 0.001 0.013 0.001 0.001 0.002

1.068 1.250 1.074 1.681 1.074 1.250 1.068

0.95 4.01 6.53 4.01 0.95

170.58 136.86 113.42 136.86 170.58


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 AADa a

cc-pVTZ | ERROR|

EOMIPCCSD(2)

99 174 222 469 567 605 710 716 775 785 1005 1070 1760 1794 3275 3275 3373 3380 32

136 217 270 551 641 652 655 711 754 756 985 1058 1800 1817 3266 3267 3385 3399

37 43 48 82 74 47 55 5 21 29 20 12 40 23 9 8 12 19

139 211 247 537 673 683 690 732 766 769 998 1076 1812 1830 3271 3272 3394 3407

|error|

1.424 1.424 1.401 1.436 1.436 1.401

1.424 1.424 1.400 1.437 1.437 1.400 0.01

0 0 0.001 0.001 0.001 0.001

1.407 1.407 1.385 1.420 1.420 1.385

bond angle (deg) C1−C2−C3 C2−C3−C4 C3−C4−C5 C4−C5−C6 C5−C6−C1 C6−C1−C2 AADb

120.92 119.72 119.76 119.77 119.76 119.72

121.04 119.66 119.80 119.89 119.80 119.66 0.07

0.12 0.06 0.04 0.12 0.04 0.06

121.20 119.56 119.74 120.03 119.74 119.56

and ab initio methods provide a valuable tool for characterization of these compounds. Table 20 compares the structural parameters calculated using the EOMIP-CCSD(2) and EOMIP-CCSD methods in the ccpVDZ basis set. The average absolute deviations in bond length and bond angles are 0.003 Å and 3.39°, respectively. The bond lengths in the EOMIP-CCSD(2) method shrink, upon moving to the cc-pVTZ basis set. Table 21 shows that the average absolute deviation in the frequency is only 32 cm−1, which indicates the power of the EOMIPCCSD(2) method to reproduce the IR spectra computed using the expensive EOMIP-CCSD method, with reasonable accuracy. Benzene Dimer Cation. The benzene dimer cation has three possible structures: x-displaced structure, y-displaced structure, and T-shaped structure. We have taken the y-displaced structure as our test case. Table 22 shows that ionization-induced structural changes using the EOMIP-CCSD method is reproduced with excellent accuracy in the EOMIP-CCSD(2) method, which is indicated by small average absolute deviations in the bond length (0.001 Å) and bond angles (0.07°) in the 6-31G basis set. The bond lengths shrink, using the EOMIP-CCSD(2) method, upon moving to the 6-31G* basis set. Table 23 shows that the IR frequencies computed using the EOMIP-CCSD(2) method show excellent agreement with those in the EOMIP-CCSD method and the absolute deviation in frequency is only 16 cm−1.


EOMIPCCSD(2)

bond length (Å) C1−C2 C2−C3 C3−C4 C4−C5 C5−C6 C6−C1 AADa

6-31G* EOMIPCCSD(2)

a

Table 21. Harmonic Vibrational Frequencies of Acetylene Dimer Cation cc-pVDZ

EOMIPCCSD(2)


a

EOMIPCCSD

EOMIPCCSD


4. CONCLUSIONS We have presented a benchmark study on the performance of the EOMIP-CCSD(2) method for geometry and vibrational frequencies of doublet radicals. The method, being naturally spin-adapted and equipped to address multireference situations, can avoid the problems associated with the standard singlereference ab initio treatment of open-shell radicals. In addition to

Acetylene Dimer Cation. Acetylene dimer cations are believed to be the building blocks of large polycyclic aromatic hydrocarbon in interstellar clouds. High-energy interstellar conditions make the scope of the experiments very limited, 4326



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Table 23. Harmonic Vibrational Frequencies of the y-Displaced Benzene Dimer Cation Harmonic Vibrational Frequencies (cm−1) cc-pVDZ EOMIP-CCSD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

a

29 46 66 76 93 178 370 372 373 384 546 554 599 601 607 623 703 712 822 827 856 866 905 912 928 930 938 941 969 988 1030 1032 1038


cc-pVTZ

cc-pVDZ

EOMIP-CCSD(2) |ERROR| EOMIP-CCSD(2) 32 47 73 79 101 185 366 367 371 383 521 536 601 602 608 624 704 714 818 823 859 862 898 899 904 906 928 929 968 985 1032 1034 1046

2 0 8 3 8 7 4 5 2 1 24 18 2 1 1 1 1 3 4 3 4 4 7 13 24 24 10 12 1 4 1 2 7

EOMIP-CCSD 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 AADa

34 51 69 82 109 200 367 368 375 390 522 540 586 589 598 614 716 726 835 839 882 892 923 930 937 939 956 958 995 1013 1022 1025 1053

1039 1063 1065 1192 1204 1226 1227 1228 1237 1337 1350 1412 1412 1503 1504 1504 1515 1554 1558 1589 1644 3159 3160 3169 3169 3170 3173 3180 3182 3188 3189 3195 3196

cc-pVTZ

EOMIP-CCSD(2) |ERROR| EOMIP-CCSD(2) 1046 1067 1069 1205 1214 1237 1237 1240 1247 1417 1420 1420 1442 1502 1505 1506 1512 1553 1555 1584 1634 3200 3201 3214 3214 3214 3217 3223 3226 3232 3232 3238 3240 16

7 4 4 13 10 12 10 12 10 80 70 8 29 2 1 2 4 0 3 5 11 41 41 45 45 44 44 43 44 43 43 43 43

1055 1086 1087 1207 1214 1221 1225 1230 1236 1396 1397 1492 1513 1514 1522 1523 1541 1592 1593 1625 1680 3248 3249 3260 3261 3262 3263 3267 3270 3274 3275 3278 3278


absolute deviation in bond length (|Δre| = 0.010 Å), as well as in IR frequency (|Δωe| = 111 cm−1). The method performs similar to that of the standard EOMIP-CCSD method, even better in some particular cases, despite the latter being more computationally expensive. The UHF reference-based MBPT(2) and CCSD methods fail to reproduce even the qualitative trends for most of the cases, which is indicated by the high value of the maximum and average absolute deviations in bond length and IR frequency. However, the ROHF-based CCSD and MBPT(2) method shows comparatively better results than the UHF-based CCSD and MBPT(2) method, but performs poorly compared to the EOMIP-CCSD(2) method. Inclusion of partial triples will obviously improve the results for the single-reference coupled-cluster methods. However, it will also increase the scaling to N7, which will make the method computationally unfeasible, even for the moderately sized molecules. Our calculations on the larger radicals and dimer cations show that the geometries and IR frequencies, obtained using the EOMIP-CCSD method, are well-reproduced in the lessexpensive EOMIP-CCSD(2) method . The EOMIP-CCSD(2) method offers an efficient black box approach for the theoretical treatment of doublet radicals and

that, the method is computationally less expensive than the standard EOMIP-CCSD and single-reference coupled-cluster methods, in terms of both computational scaling as well as storage requirements. The performance of the method is benchmarked, in a hierarchy of Dunning’s correlation-consistent aug-cc-pVXZ (X = D, T, Q) basis sets, on a variety of doublet radicals, which are previously reported to be challenging cases for standard ab initio methods. We have compared our results with the EOMIP-CCSD method, the UHF- and ROHF-based coupled-cluster methods, and MBPT(2) method. The computed results demonstrate that the EOMIP-CCSD(2) method provides reasonable agreement with the experimental geometry and IR frequency. The calculated bond lengths and frequencies are strongly dependent on the basis sets used. The bond lengths decrease and IR frequencies shift to higher values upon changing the basis set from the aug-cc-pVDZ basis set to the aug-cc-pVTZ basis set. The change in the results is negligible upon moving from the aug-cc-pVTZ basis set to the aug-cc-pVQZ basis set; hence, the results in the aug-cc-pVQZ basis set can be taken as the complete basis set limit results. We have calculated the minimum, maximum, and average absolute deviations from the experiment for all of the methods, in the aug-cc-pVQZ basis set. The EOMIP-CCSD(2) method shows the smallest average 4327



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gives accuracy comparable to the robust EOMIP-CCSD method, at significantly lower computational cost. The method has immense potential to be used in the theoretical treatment of open-shell radicals, biological molecules, and gasphase clusters. Current EOM-IP-CCSD codes can be used to describe systems as large as a nucleobase tetramer;66,67 therefore, cutting the scaling can really extend the range of systems. Work is currently underway to study the properties and thermochemistry of DNA base pairs and ionized water clusters.

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ASSOCIATED CONTENT

S Supporting Information *

Z-Matrix of the optimized geometries, nuclear repulsion energy, SCF energy, correlation energy, final energy, vibrational frequency, and infrared intensity are given as Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge a grant from the CSIR XIIth Five Year Plan Project on Multiscale Simulations of Material (MSM) and facilities of the Centre of Excellence in Scientific Computing at NCL. A.K.D. thanks the Council of Scientific and Industrial Research (CSIR) for a Senior Research Fellowship. S.P. acknowledges the DST J. C. Bose Fellowship project and a CSIR SSB grant towards completion of the work.

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